These notes come from courses and seminars given by the authors at the University of Salento and at the University of Napoli "Federico II", and from several conversations between the authors themselves on a theory - that of groups with finite conjugacy classes - that seventy years after its birth remains vital and full of problems still to be solved. There are two relevant monographs on the theory of FC-groups, one due to Y.M. Garcakov and the order to M.J. Tomkinson, but both of them go back to over thirty years ago, and since then many new contributions have brought new light to the existing problems and posed new relevant questions. Our treatment has no claim to completeness, but is intended to provide the essential tools to the theory and some of its recent developments, suggesting possible further research topics.
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